11주차

Introduction to Probability and Statistics
11th Week (5/24)

Hypothesis Testing

Hypothesis
in statistics, is a claim or statement about
a property of a population

Hypothesis Testing
is to test the claim or statement
Example: A conjecture is made that “the
average starting salary for computer
science gradate is $30,000 per year”.

Nonstatistical Hypothesis Testing…
A criminal trial is an example of hypothesis testing without
the statistics.
In a trial a jury must decide between two hypotheses. The
null hypothesis is
H0: The defendant is innocent

The alternative hypothesis or research hypothesis is
H1: The defendant is guilty

The jury does not know which hypothesis is true. They must
make a decision on the basis of evidence presented.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.3

Nonstatistical Hypothesis Testing…
In the language of statistics convicting the defendant is called rejecting
the null hypothesis in favor of the alternative hypothesis. That is, the
jury is saying that there is enough evidence to conclude that the
defendant is guilty (i.e., there is enough evidence to support the
alternative hypothesis).

If the jury acquits it is stating that there is not enough evidence to
support the alternative hypothesis. Notice that the jury is not saying
that the defendant is innocent, only that there is not enough evidence to
support the alternative hypothesis. That is why we never say that we
accept the null hypothesis, although most people in industry will say
“We accept the null hypothesis”


Question:
How can we justify/test this conjecture?
A. What do we need to know to justify
this conjecture?
B. Based on what we know, how should
we justify this conjecture?

Answer to A:
Randomly select, say 100, computer
science graduates and find out their
annual salaries
---- We need to have some sample
observations, i.e., a sample set!

Answer to B:
That is what we will learn in this
chapter
---- Make conclusions based on the
sample observations

Statistical Reasoning
Analyze the sample set in an attempt to
distinguish between results that can
easily occur and results that are highly
unlikely.

Statistical Decisions

Decisions about populations on the basis of sample information.
Ex) We may wish to decide on the basis of sample data whether a new
serum is really effective in curing a disease, or whether one educational
procedure is better than another

Definitions
 Null Hypothesis (denoted H 0):
is the statement being tested in a
test of hypothesis.

 Alternative Hypothesis (H 1):
is what is believe to be true if the
null hypothesis is false.

Null Hypothesis: H0
 Must contain condition of equality
 =, ≥, or ≤
 Test the Null Hypothesis directly

 Reject H 0 or fail to reject H 0

Alternative Hypothesis: H1
 Must be true if H0 is false

 ≠, <, >
 ‘opposite’ of Null
Example:

H0 : µ = 30 versus H1 : µ > 30

Statistical Hypotheses and
Null Hypotheses
•Statistical hypotheses: Assumptions or guesses about the populations
involved. (Such assumptions, which may or may not be true)

•Null hypotheses (H0): Hypothesis that there is no difference between the
procedures. We formulate it if we want to decide whether one procedure is
better than another.

•Alternative hypotheses (H1): Any hypothesis that differs from a given null
hypothesis

Example 1. For example, if the null hypothesis is p = 0.5, possible
alternative hypotheses are p =0.7, or p ≠ 0.5.

Concepts of Hypothesis Testing
(1)…
• The two hypotheses are called the null hypothesis and
the other the alternative or research hypothesis. The
usual notation is:
pronounced
H “nought”

• H0: — the ‘null’ hypothesis

• H1: — the ‘alternative’ or ‘research’ hypothesis

• The null hypothesis (H0) will always state that the
parameter equals the value specified in the alternative
11.14
hypothesis (H1)

Stating Your Own Hypothesis
If you wish to support your claim, the
claim must be stated so that it becomes
the alternative hypothesis.

Important Notes:
H0 must always contain equality; however some
claims are not stated using equality. Therefore
sometimes the claim and H0 will not be the
same.

Ideally all claims should be stated that they are
Null Hypothesis so that the most serious error
would be a Type I error.

Tests of Hypotheses and Significance

“Significant”: If on the supposition that a particular hypothesis is true we
find that results observed in a random sample differ markedly from those
expected under the hypothesis on the basis of pure chance using
sampling theory, we would say that the observed differences are
significant

•We would be inclined to reject the hypothesis if the observed differences
are significant.

• Tests of hypotheses, tests of significance, or decision rules: Procedures
that enable us to decide whether to accept or reject hypotheses or to
determine whether observed samples differ significantly from expected
results

Type I Error
The mistake of rejecting the null hypothesis
when it is true.

The probability of doing this is called the
significance level, denoted by α (alpha).

Common choices for α: 0.05 and 0.01

Example: rejecting a perfectly good parachute
and refusing to jump

Type II Error
the mistake of failing to reject the null
hypothesis when it is false.

denoted by ß (beta)

Example: failing to reject a defective
parachute and jumping out of a
plane with it.

Table 7-2 Type I and Type II Errors
True State of Nature
The null The null
hypothesis is hypothesis is
true false
We decide to Type I error
Correct
reject the (rejecting a true
decision
null hypothesis null hypothesis)
Decision
We fail to Type II error
Correct
reject the (failing to reject
decision
null hypothesis a false null
hypothesis)

Types of Errors…
A Type I error occurs when we reject a true null hypothesis
(i.e. Reject H0 when it is TRUE)

H0 T F

Reject I

Reject II

A Type II error occurs when we don’t reject a false null
hypothesis (i.e. Do NOT reject H0 when it is FALSE)

Type of Errors…
There are two possible errors.
A Type I error occurs when we reject a true null hypothesis.
That is, a Type I error occurs when the jury convicts an
innocent person. We would want the probability of this type
of error [maybe 0.001 – beyond a reasonable doubt] to be
very small for a criminal trial where a conviction results in
the death penalty, whereas for a civil trial, where conviction
might result in someone having to “pay for damages to a
wrecked auto”,we would be willing for the probability to be
larger [0.49 – preponderance of the evidence ]
P(Type I error) = α [usually 0.05 or 0.01]


Type of Errors…
A Type II error occurs when we don’t reject a false null
hypothesis [accept the null hypothesis]. That occurs when a
guilty defendant is acquitted.
In practice, this type of error is by far the most serious
mistake we normally make. For example, if we test the
hypothesis that the amount of medication in a heart pill is
equal to a value which will cure your heart problem and
“accept the hull hypothesis that the amount is ok”. Later on
we find out that the average amount is WAY too large and
people die from “too much medication” [I wish we had
rejected the hypothesis and threw the pills in the trash can],
it’s too late because we shipped the pills to the public.

Type of Errors…
The probability of a Type I error is denoted as α (Greek letter
alpha). The probability of a type II error is β (Greek letter
beta).

The two probabilities are inversely related. Decreasing one
increases the other, for a fixed sample size.

In other words, you can’t have α and β both real small for
any old sample size. You may have to take a much larger
sample size, or in the court example, you need much more
evidence.


Type of Errors…
The critical concepts are theses:
1. There are two hypotheses, the null and the alternative hypotheses.
2. The procedure begins with the assumption that the null hypothesis is
true.
3. The goal is to determine whether there is enough evidence to infer
that the alternative hypothesis is true, or the null is not likely to be
true.
4. There are two possible decisions:
Conclude that there is enough evidence to support the alternative
hypothesis. Reject the null.
Conclude that there is not enough evidence to support the
alternative hypothesis. Fail to reject the null.


Judging the Test…
A statistical test of hypothesis is effectively defined by the
significance level ( ) and the sample size (n), both of
which are selected by the statistics practitioner.

Therefore, if the probability of a Type II error ( ) is too
large [we have insufficient power], we can reduce it by
increasing , and/or
increasing the sample size, n.


Judging the Test…
The power of a test is defined as 1– .
It represents the probability of rejecting the null hypothesis when it is
false and the true mean is something other than the null value for the
mean.

If we are testing the hypothesis that the average amount of medication
in blood pressure pills is equal to 6 mg (which is good), and we “fail to
reject” the null hypothesis, ship the pills to patients worldwide, only to
find out later that the “true” average amount of medication is really 8
mg and people die, we get in trouble. This occurred because the
P(reject the null / true mean = 7 mg) = 0.32 which would mean that we
have a 68% chance on not rejecting the null for these BAD pills and
shipping to patients worldwide.


Type I and Type II Errors

•Type I error: If we reject a hypothesis when it happens to be true.

•Type II error: If we accept a hypothesis when it should be rejected.

•In order for any tests of hypotheses or decision rules to be good, they
must be designed so as to minimize errors of decision.

• An attempt to decrease one type of error is accompanied in general by an
increase in the other type of error. The only way to reduce both types of
error is to increase the sample size, which may or may not be possible.

Significant Differences
 Hypothesis testing is designed to detect
significant differences: differences that did not
occur by random chance.

 In the “one sample” case: we compare a random
sample (from a large group) to a population.

 We compare a sample statistic to a population
parameter to see if there is a significant
difference.

Level of Significance ( 유의수준 )

• Level of significance: In testing a given hypothesis, the maximum
probability with which we would be willing to risk a Type I error is called the
level of significance

Level of Significance
•In practice a level of significance of 0.05 or 0.01 is customary, although
other values are used.

• If for example a 0.05 or 5% level of significance is chosen in designing a
test of a hypothesis, then there are about 5 chances in 100 that we would
reject the hypothesis when it should be accepted, i.e., whenever the null
hypotheses is true, we are about 95% confident that we would make the
right decision. In such cases we say that the hypothesis has been rejected
at a 0.05 level of significance, which means that we could be wrong with
probability 0.05.

Definition
Test Statistic:
is a sample statistic or value based
on sample data

Example:

x – µx
z =
σ/ n

Definition
Critical Region :
is the set of all values of the test statistic
that would cause a rejection of the null
hypothesis

Critical Region
• Set of all values of the test statistic that
would cause a rejection of the
null hypothesis

Critical
Region

Critical Region
• null hypothesis
Critical
Region

Critical Region
null hypothesis
Critical
Regions

Definition
Critical Value:
is the value (s) that separates the critical
region from the values that would not lead
to a rejection of H 0

Critical Value
Value (s) that separates the critical region
from the values that would not lead to a
rejection of H 0

Critical Value
( z score )

Critical Value
Value (s) that separates the critical region
from the values that would not lead to a
rejection of H 0

Reject H0 Fail to reject H0

Critical Value
( z score )

Tests Involving the Normal Distribution
- Level of confidence : 0.05

•The critical region (or region of rejection of the hypothesis or the region
of significance): The set of z scores outside the range -1.96 to 1.96
constitutes

• The region of acceptance of the hypothesis (or the region of
nonsignificance) : The set of z scores inside the range -1.96 to 1.96 could

Tests Involving the Normal Distribution
• Decision Rule

• When the level of confidence is 0.01, a value 2.58 should be instead of 1.96.

Two-tailed,
Left-tailed,
Right-tailed
Tests

Left-tailed Test
H0: µ ≥ 200

H1: µ < 200

Left-tailed Test
H0: µ ≥ 200

H1: µ < 200
Points Left

Left-tailed Test
H0: µ ≥ 200

H1: µ < 200
Points Left

Reject H0 Fail to reject H0

Values that
differ significantly
from 200 200

Right-tailed Test
H0: µ ≤ 200

H1: µ > 200

Right-tailed Test
H0: µ ≤ 200

H1: µ > 200
Points Right

Right-tailed Test
H0: µ ≤ 200

H1: µ > 200
Points Right

Fail to reject H0 Reject H0

Values that
differ significantly
200 from 200

Two-tailed Test
H0: µ = 200
H1: µ ≠ 200

Two-tailed Test
H0: µ = 200 α is divided equally between
the two tails of the critical
H1: µ ≠ 200 region

Two-tailed Test

Means less than or greater than

Two-tailed Test

Means less than or greater than

Reject H0 Fail to reject H0 Reject H0

200

Values that differ significantly from 200

Summary of One- and Two-Tail Tests…

One-Tail Test Two-Tail Test One-Tail Test
(left tail) (right tail)


One-Tailed and Two-Tailed Tests
•Two-tailed tests or two-sided tests: When we display interest in extreme
values of the statistic S or its corresponding z score on both sides of the
mean, i.e., in both tails of the distribution.

• One-tailed tests or one-sided tests: When we are interested only in
extreme values to one side of the mean, i.e., in one tail of the distribution,
as, for example, when we are testing the hypothesis that one process is
better than another (which is different from testing whether one process is
better or worse than the other).

P Value

• The null hypothesis H0 will be an assertion that a population
parameter has a specific value, and the alternative hypothesis H1 will be
one of the following assertions:

(i) The parameter is greater than the stated value (right-tailed test).
(ii) The parameter is less than the stated value (left-tailed test).
(iii) The parameter is either greater than or less than the stated value (two-
tailed test).

• P value of the test: The probability that a value of S in the direction(s) of
H1 and as extreme as the one that actually did occur would occur if H0
were true.

Interpreting the p-value…
The smaller the p-value, the more statistical evidence exists
to support the alternative hypothesis.
•If the p-value is less than 1%, there is overwhelming
evidence that supports the alternative hypothesis.
•If the p-value is between 1% and 5%, there is a strong
•If the p-value is between 5% and 10% there is a weak
•If the p-value exceeds 10%, there is no evidence that
supports the alternative hypothesis.
We observe a p-value of .0069, hence there is
overwhelming evidence to support H1: > 170.


Interpreting the p-value…
Overwhelming Evidence
(Highly Significant)
Strong Evidence
(Significant)

Weak Evidence
(Not Significant)

No Evidence
(Not Significant)

0 .01 .05 .10

p=.0069

P Value

•Small P values provide evidence for rejecting the null hypothesis in favor of
the alternative hypothesis, and large P values provide evidence for not
rejecting the null hypothesis in favor of the alternative hypothesis.

•The P value and the level of significance do not provide criteria for
rejecting or not rejecting the null hypothesis by itself, but for rejecting or
not rejecting the null hypothesis in favor of the alternative hypothesis.

• When the test statistic S is the standard normal random variable, the
table in Appendix C is sufficient to compute the P value, but when S is one
of the t, F, or chi-square random variables, all of which have different
distributions depending on their degrees of freedom, either computer
software or more extensive tables will be needed to compute the P value.

Special Tests of Significance for Large
Samples: Means

Our Problem:
 The education department at a university has been
accused of “grade inflation” so education majors
have much higher GPAs than students in general.

 GPAs of all education majors should be compared
with the GPAs of all students.
 There are 1000s of education majors, far too many to
interview.
 How can this be investigated without interviewing all
education majors?

What we know:
 The average GPA for
all students is 2.70. µ = 2.70
This value is a
parameter.

 To the right is the
X = 3.00
statistical information
for a random sample
s= 0.70
of education majors:
N= 117

Questions to ask:

 Is there a difference between the parameter
(2.70) and the statistic (3.00)?

 Could the observed difference have been
caused by random chance?

 Is the difference real (significant)?

Two Possibilities:

1. The sample mean (3.00) is the same as
the pop. mean (2.70).
 The difference is trivial and caused by
random chance.

1. The difference is real (significant).
 Education majors are different from all
students.

The Null and Alternative Hypotheses:
1. Null Hypothesis (H0)
 The difference is caused by random chance.
 The H0 always states there is “no significant difference.” In
this case, we mean that there is no significant difference
between the population mean and the sample mean.
1. Alternative hypothesis (H1)
 “The difference is real”.
 (H1) always contradicts the H0.

 One (and only one) of these explanations must be true.
Which one?

Test the Explanations
 We always test the Null Hypothesis.
 Assuming that the H0 is true:
 What is the probability of getting the sample
mean (3.00) if the H0 is true and all education
majors really have a mean of 2.70? In other
words, the difference between the means is
due to random chance.
 If the probability associated with this difference
is less than 0.05, reject the null hypothesis.

Test the Hypotheses
 Use the .05 value as a guideline to identify differences
that would be rare or extremely unlikely if H0 is true.
This “alpha” value delineates the “region of rejection.”

 Use the Z score formula for single samples and
Appendix A to determine the probability of getting the
observed difference.

 If the probability is less than .05, the calculated or
“observed” Z score will be beyond ±1.96 (the “critical”
Z score).

Two-tailed Hypothesis Test:

Z= -1.96 Z = +1.96

c c

When α = .05, then .025 of the area is distributed on either
side of the curve in area (C )
The .95 in the middle section represents no significant
difference between the population and the sample mean.
The cut-off between the middle section and +/- .025 is
represented by a Z-value of +/- 1.96.

Testing Hypotheses:
Using The Five Step Model…
1. Make Assumptions and meet test
requirements.
2. State the null hypothesis.
3. Select the sampling distribution and
establish the critical region.
4. Compute the test statistic.
5. Make a decision and interpret results.

Step 1: Make Assumptions and Meet
Test Requirements
 Random sampling
 Hypothesis testing assumes samples were selected using
random sampling.
 In this case, the sample of 117 cases was randomly selected
from all education majors.

 Level of Measurement is Interval-Ratio
 GPA is I-R so the mean is an appropriate statistic.

 Sampling Distribution is normal in shape
 This is a “large” sample (N≥100).

Step 2 State the Null Hypothesis

 H0: μ = 2.7 (in other words, H0: = μ)
 You can also state Ho: No difference between the sample
mean and the population parameter
 (In other words, the sample mean of 3.0 really the same as
the population mean of 2.7 – the difference is not real but
is due to chance.)
 The sample of 117 comes from a population that has a
GPA of 2.7.
 The difference between 2.7 and 3.0 is trivial and caused by
random chance.

Step 2 (cont.) State the Alternate Hypothesis
 H1: μ≠2.7 (or, H0: ≠ μ)
 Or H1: There is a difference between the sample mean and
the population parameter
 The sample of 117 comes a population that does not have
a GPA of 2.7. In reality, it comes from a different population.
 The difference between 2.7 and 3.0 reflects an actual
difference between education majors and other students.
 Note that we are testing whether the population the sample
comes from is from a different population or is the same as
the general student population.

Step 3 Select Sampling Distribution and
Establish the Critical Region
 Sampling Distribution= Z

 Alpha (α) = .05

 α is the indicator of “rare” events.

 Any difference with a probability less than α
is rare and will cause us to reject the H 0.

Step 3 (cont.) Select Sampling Distribution
and Establish the Critical Region
 Critical Region begins at Z= ± 1.96

 This is the critical Z score associated
with α = .05, two-tailed test.

 If the obtained Z score falls in the Critical
Region, or “the region of rejection,” then
we would reject the H0.

Step 4: Use Formula to Compute the Test
Statistic (Z for large samples (≥ 100)

Χ− µ
Z=
σ N

When the Population σ is not known,
use the following formula:

Χ−µ
Z=
s N −1

Test the Hypotheses
3.0 − 2.7
Z= = 4.62
.7
117 − 1
 We can substitute the sample standard deviation
S for σ (pop. s.d.) and correct for bias by
substituting N-1 in the denominator.
 Substituting the values into the formula, we
calculate a Z score of 4.62.

Step 5 Make a Decision and Interpret
Results
 The obtained Z score fell in the Critical Region, so we reject
the H0.
 If the H0 were true, a sample outcome of 3.00 would be
unlikely.
 Therefore, the H0 is false and must be rejected.

 Education majors have a GPA that is significantly different
from the general student body (Z = 4.62, α = .05).*

 *Note: Always report significant statistics.

Looking at the curve:
(Area C = Critical Region when α=.05)

Z= -1.96 Z = +1.96

c c z= +4.62
I

Summary:
 The GPA of education majors is significantly
different from the GPA of the general student body.

 In hypothesis testing, we try to identify statistically
significant differences that did not occur by random
chance.

 In this example, the difference between the
parameter 2.70 and the statistic 3.00 was large and
unlikely (p < .05) to have occurred by random
chance.

Summary (cont.)

 We rejected the H0 and concluded that the
difference was significant.

 It is very likely that Education majors have
GPAs higher than the general student body

Samples: Proportions

Samples: Difference of Means

Samples: Difference of Proportions

Special Tests of Significance for Small
Samples: Means

Using the Student’s t Distribution for
Small Samples (One Sample T-Test)
 When the sample size is small
(approximately < 100) then the Student’s t
distribution should be used (see Appendix B)
 The test statistic is known as “t”.
 The curve of the t distribution is flatter than
that of the Z distribution but as the sample
size increases, the t-curve starts to resemble
the Z-curve (see text p. 230 for illustration)

Degrees of Freedom

 The curve of the t distribution varies with
sample size (the smaller the size, the flatter
the curve)
 In using the t-table, we use “degrees of
freedom” based on the sample size.
 For a one-sample test, df = N – 1.
 When looking at the table, find the t-value for
the appropriate df = N-1. This will be the
cutoff point for your critical region.

Formula for one sample t-test:

Χ−µ
t=
S
N −1

Example

 A random sample of 26 sociology
graduates scored 458 on the GRE
advanced sociology test with a standard
deviation of 20. Is this significantly
different from the population average
(µ = 440)?

Solution (using five step model)

 Step 1: Make Assumptions and Meet Test
Requirements:

 1. Random sample
 2. Level of measurement is interval-ratio
 3. The sample is small (<100)

Solution (cont.)

Step 2: State the null and alternate hypotheses.

H0: µ = 440 (or H0: = μ)

H1: µ ≠ 440

Solution (cont.)
 Step 3: Select Sampling Distribution and
Establish the Critical Region

1. Small sample, I-R level, so use t
distribution.
2. Alpha (α) = .05
3. Degrees of Freedom = N-1 = 26-1 = 25
4. Critical t = ±2.060

Solution (cont.)
 Step 4: Use Formula to Compute the Test Statistic

Χ−µ
458 − 440
t= = = 4.5
S 20
N −1 26 − 1

Looking at the curve for the t distribution
Alpha (α) = .05

t= -2.060 t = +2.060

c c t= +4.50
I

Step 5 Make a Decision and Interpret
Results
 The obtained t score fell in the Critical Region, so
we reject the H0 (t (obtained) > t (critical)
 If the H0 were true, a sample outcome of 458
would be unlikely.
 Therefore, the H0 is false and must be rejected.

 Sociology graduates have a GRE score that is
significantly different from the general student body
(t = 4.5, df = 25, α = .05).

Testing Sample Proportions:

 When your variable is at the nominal (or
ordinal) level the one sample z-test for
proportions should be used.
 If the data are in % format, convert to a
proportion first.
 The method is the same as the one sample
Z-test for means (see above)

Samples: Variance

Samples: Difference of Means

Samples: Ratios of Variances

Concepts of Hypothesis Testing…
For example, if we’re trying to decide whether the mean is
not equal to 350, a large value of (say, 600) would provide
enough evidence.

If is close to 350 (say, 355) we could not say that this
provides a great deal of evidence to infer that the population
mean is different than 350.


Concepts of Hypothesis Testing (4)…
The two possible decisions that can be made:

Conclude that there is enough evidence to support the alternative
hypothesis
(also stated as: reject the null hypothesis in favor of the alternative)

Conclude that there is not enough evidence to support the
alternative hypothesis
(also stated as: failing to reject the null hypothesis in favor of the
alternative)
NOTE: we do not say that we accept the null hypothesis if a
statistician is around…


Concepts of Hypothesis Testing (2)…
The testing procedure begins with the assumption that the
null hypothesis is true.

Thus, until we have further statistical evidence, we will
assume:

H0: = 350 (assumed to be TRUE)
The next step will be to determine the sampling distribution
of the sample mean assuming the true mean is 350.
is normal with 350
75/SQRT(25) = 15

Is the Sample Mean in the Guts of the Sampling Distribution??


Three ways to determine this: First way
1. Unstandardized test statistic: Is in the guts of the
sampling distribution? Depends on what you define as
the “guts” of the sampling distribution.

If we define the guts as the center 95% of the distribution
[this means α = 0.05], then the critical values that define
the guts will be 1.96 standard deviations of X-Bar on
either side of the mean of the sampling distribution
[350], or
UCV = 350 + 1.96*15 = 350 + 29.4 = 379.4
LCV = 350 – 1.96*15 = 350 – 29.4 = 320.6


1. Unstandardized Test Statistic Approach


Three ways to determine this: Second way
2. Standardized test statistic: Since we defined the “guts” of
the sampling distribution to be the center 95% [α = 0.05],
If the Z-Score for the sample mean is greater than
1.96, we know that will be in the reject region on the right
side or
If the Z-Score for the sample mean is less than -1.97,
we know that will be in the reject region on the left side.

Z=( - )/ = (370.16 – 350)/15 = 1.344

Is this Z-Score in the guts of the sampling distribution???

2. Standardized Test Statistic Approach


Three ways to determine this: Third way
3. The p-value approach (which is generally used with a computer and
statistical software): Increase the “Rejection Region” until it
“captures” the sample mean.

For this example, since is to the right of the mean, calculate
P( > 370.16) = P(Z > 1.344) = 0.0901
Since this is a two tailed test, you must double this area for the p-value.
p-value = 2*(0.0901) = 0.1802
Since we defined the guts as the center 95% [α = 0.05], the reject
region is the other 5%. Since our sample mean, , is in the 18.02%
region, it cannot be in our 5% rejection region [α = 0.05].


3. p-value approach


Statistical Conclusions:
Unstandardized Test Statistic:
Since LCV (320.6) < (370.16) < UCV (379.4), we
reject the null hypothesis at a 5% level of significance.

Standardized Test Statistic:
Since -Zα/2(-1.96) < Z(1.344) < Zα/2 (1.96), we fail to reject
the null hypothesis at a 5% level of significance.

P-value:
Since p-value (0.1802) > 0.05 [α], we fail to reject the
hull hypothesis at a 5% level of significance.

Example 11.1…
A department store manager determines that a new billing
system will be cost-effective only if the mean monthly
account is more than $170.

A random sample of 400 monthly accounts is drawn, for
which the sample mean is $178. The accounts are
approximately normally distributed with a standard deviation
of $65.

Can we conclude that the new system will be cost-effective?


Example 11.1…
The system will be cost effective if the mean account
balance for all customers is greater than $170.

We express this belief as a our research hypothesis, that is:

H1: > 170 (this is what we want to determine)

Thus, our null hypothesis becomes:

H0: = 170 (this specifies a single value for the
parameter of interest) – Actually H0: μ < 170

Example 11.1…
What we want to show:
H1: > 170
H0: < 170 (we’ll assume this is true)
Normally we put Ho first.
We know:
n = 400,
= 178, and
= 65
= 65/SQRT(400) = 3.25
α = 0.05

Example 11.1… Rejection Region…
The rejection region is a range of values such that if the test
statistic falls into that range, we decide to reject the null
hypothesis in favor of the alternative hypothesis.

is the critical value of to reject H0.

Example 11.1…
At a 5% significance level (i.e. =0.05), we get [all α in one tail]
Zα = Z0.05 = 1.645
Therefore, UCV = 170 + 1.645*3.25 = 175.35
Since our sample mean (178) is greater than the critical value we
calculated (175.35), we reject the null hypothesis in favor of H1
OR
(>1.645) Reject null

OR
p-value = P( > 178) = P(Z > 2.46) = 0.0069 < 0.05 Reject null


Example 11.1… The Big Picture…

H1: > 170 =175.34
H0: = 170
=178
Reject H0 in favor of

Conclusions of a Test of Hypothesis…
If we reject the null hypothesis, we conclude that there is
enough evidence to infer that the alternative hypothesis is
true.

If we fail to reject the null hypothesis, we conclude that there
is not enough statistical evidence to infer that the alternative
hypothesis is true. This does not mean that we have proven
that the null hypothesis is true!

Keep in mind that committing a Type I error OR a Type II
error can be VERY bad depending on the problem.


One tail test with rejection region on right
The last example was a one tail test, because the rejection
region is located in only one tail of the sampling distribution:

More correctly, this was an example of a right tail test.
H1: μ > 170
H0: μ < 170


One tail test with rejection region on left
The rejection region will be in the left tail.


Two tail test with rejection region in both tails
The rejection region is split equally between the two tails.


Example 11.2… Students work
AT&T’s argues that its rates are such that customers won’t
see a difference in their phone bills between them and their
competitors. They calculate the mean and standard deviation
for all their customers at $17.09 and $3.87 (respectively).
Note: Don’t know the true value for σ, so we estimate σ from
the data [σ ~ s = 3.87] – large sample so don’t worry.
They then sample 100 customers at random and recalculate a
monthly phone bill based on competitor’s rates.
Our null and alternative hypotheses are
H1: ≠ 17.09. We do this by assuming that:
H0: = 17.09


Example 11.2…
The rejection region is set up so we can reject the null
hypothesis when the test statistic is large or when it is small.

stat is “small” stat is “large”

That is, we set up a two-tail rejection region. The total area
in the rejection region must sum to , so we divide α by 2.


Example 11.2…
At a 5% significance level (i.e. = .05), we have
/2 = .025. Thus, z.025 = 1.96 and our rejection region is:

z < –1.96 -or- z > 1.96

-z.025 +z.025 z
0


Example 11.2…
From the data, we calculate = 17.55

Using our standardized test statistic:

We find that:

Since z = 1.19 is not greater than 1.96, nor less than –1.96
we cannot reject the null hypothesis in favor of H1. That is
“there is insufficient evidence to infer that there is a
difference between the bills of AT&T and the competitor.”

Probability of a Type II Error –
A Type II error occurs when a false null hypothesis is not
rejected or “you accept the null when it is not true” but don’t
say it this way if a statistician is around.

In practice, this is by far the most serious error you can make
in most cases, especially in the “quality field”.


Probability you ship pills whose mean amount of medication is 7 mg approximately 67%


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